I. N. Bronshtein K. A. Semendyayev G. Musiol H. Mühlig H ANDBOOK OF M ATHEMATICS Sixth Edition 123 Handbook of Mathematics I.N. Bronshtein K.A. Semendyayev • ü Gerhard Musiol Heiner M hlig • Handbook of Mathematics Sixth Edition With799 Figuresand132Tables 123 I.N.Bronshtein(Deceased) GerhardMusiol Dresden,Sachsen K.A.Semendyayev(Deceased) Germany HeinerMühlig Dresden,Sachsen Germany ISBN978-3-662-46220-1 ISBN978-3-662-46221-8 (eBook) DOI10.1007/978-3-662-46221-8 LibraryofCongressControlNumber:2015933616 SpringerHeidelbergNewYorkDordrechtLondon ©Springer-VerlagBerlinHeidelberg2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis bookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernorthe authors or the editors give a warranty, express or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer-VerlagGmbHBerlinHeidelbergispartofSpringerScience+BusinessMedia (www.springer.com) PrefacetotheSixthEnglishEdition ThissixthEnglisheditionisbasedonthefifthEnglishedition(2007)andcorrespondstotheimproved seventh(2008),eighth(2012)andninth(2013)Germanedition. Itcontainsallthechaptersofthe mentionededitions,butinarenewed,revisedandextendedform(seealsotheprefacetothefifthEnglish edition). SpecialnewpartstobementionedherearesuchsupplementarysectionsasGeometricandCoordinate TransformationsandPlainProjectionsinChapteronGeometry,asQuaternionsandApplicationsin ChapteronLinearAlgebra,asLieGroupsandLieAlgebrasinChapteronAlgebraandDiscreteMath- ematicsandasMatlabinChapteronNumericalAnalysis.TheChapteronComputerAlgebraSystems isrestrictedtoMathematicaonlyasarepresentativeexampleforsuchsystems. ExtendedandrevisedparagraphsaregiveninChapteronGeometryaboutCardanandEulerangles andaboutCoordinatetransformations,inChapteronIntegralCalculusaboutApplicationsofDefinite Integrals,inChapteronOptimizationaboutEvaluationStrategies,inChapteronTablesaboutNatural ConstantsandPhysicalUnits(SystemSI).TheIndexhasbeencompletedtoanextentasintheprevious Germaneditions. Wewouldliketocordiallythankallreadersandprofessionalcolleagueswhohelpeduswiththeirvalu- ablestatements,remarksandsuggestionsontheGermaneditionsofthebookduringtherevisionpro- cess. SpecialthanksgotoMrs. ProfessorDr. GabriellaSz´ep(Budapest), whomadethisEnglish editionpossiblebyvaluablecontributionsandthebasictranslationintotheEnglish.Furthermoreour thanksgotoallco-authorsforthecriticaltreatmentoftheirchapters. Dresden,December2014 Prof.Dr.GerhardMusiol Prof.Dr.HeinerMu¨hlig PrefacetotheFifthEnglishEdition ThisfiftheditionisbasedonthefourthEnglishedition(2003)andcorrespondstotheimprovedsixth Germanedition(2005). Itcontainsallthechaptersofthebothmentionededitions,butinarenewed revisedandextendedform. Sointheworkathand,theclassicalareasofEngineeringMathematicsrequiredforcurrentpracticeare presented,suchas“Arithmetic”,“Functions”,“Geometry”,“LinearAlgebra”,“AlgebraandDiscrete Mathematics”,(including“Logic”,“SetTheory”,“ClassicalAlgebraicStructures”,“FiniteFields”, “ElementaryNumberTheory”,”Cryptology”,“UniversalAlgebra”,“BooleanAlgebraandSwitchAl- gebra”,“AlgorithmsofGraphTheory”,“FuzzyLogic”),“Differentiation”,“IntegralCalculus”,“Dif- ferentialEquations”,“CalculusofVariations”,“LinearIntegralEquations”,“FunctionalAnalysis”, “VectorAnalysisandVectorFields”,“FunctionTheory”,“IntegralTransformations”,“Probability TheoryandMathematicalStatistics”. Fieldsofmathematicsthathavegainedimportancewithregardstotheincreasingmathematicalmod- eling and penetration of technical and scientific processes also receive special attention. Included amongstthesechaptersare“StochasticProcessesandStochasticChains”aswellas“CalculusofEr- rors”, “Dynamical Systems and Chaos”, “Optimization”, “Numerical Analysis”, “Using the Com- puter”and“ComputerAlgebraSystems”. TheChapter21containingalargenumberofusefultablesforpracticalworkhasbeencompletedby VI addingtableswiththephysicalunitsoftheInternationalSystemofUnits(SI). Dresden,February2007 Prof.Dr.GerhardMusiol Prof.Dr.HeinerMu¨hlig FromthePrefacetotheFourthEnglishEdition The“HandbookofMathematics”bythemathematician,I.N.Bronshteinandtheengineer,K.A. Semendyayevwasdesignedforengineersandstudentsoftechnicaluniversities. Itappearedforthe firsttimeinRussianandwaswidelydistributedbothasareferencebookandasatextbookforcolleges anduniversities.ItwaslatertranslatedintoGermanandthemanyeditionshavemadeitapermanent fixtureinGerman-speakingcountries,wheregenerationsofengineers,naturalscientistsandothersin technicaltrainingoralreadyworkingwithapplicationsofmathematicshaveusedit. OnbehalfofthepublishinghouseHarriDeutsch,arevisionandasubstantiallyenlargededitionwas preparedin1992byGerhardMusiolandHeinerMu¨hlig,withthegoalofgiving”Bronshtein”themod- ernpracticalcoveragerequestedbynumerousstudents, universityteachersandpractitioners. The originalstylesuccessfullyusedbytheauthorshasbeenmaintained.Itcanbecharacterizedas“short, easilyunderstandable,comfortabletouse,butfeaturingmathematicalaccuracy(atalevelofdetail consistentwiththeneedsofengineers)”∗.Since2000,therevisedandextendedfifthGermaneditionof therevisionhasbeenonthemarket. Acknowledgingthesuccessthat“Bronstein”hasexperienced intheGerman-speakingcountries,SpringerVerlagHeidelberg/GermanyispublishingafourthEnglish edition,whichcorrespondstotheimprovedandextendedfifthGermanedition. Thebookisenhancedwithoverathousandcomplementaryillustrationsandmanytables. Special functions,seriesexpansions,indefinite,definiteandellipticintegralsaswellasintegraltransformations andstatisticaldistributionsaresuppliedinanextensiveappendixoftables. Inordertomakethereferencebookmoreeffective,clarityandfastaccessthroughaclearstructure werethegoals,especiallythroughvisualcluesaswellasbyadetailedtechnicalindexandcoloredtabs. Anextendedbibliographyalsodirectsuserstofurtherresources. SpecialthanksgotoMrs.ProfessorDr.GabriellaSz´ep(Budapest),whomadethisEnglishdebutver- sionpossible. Dresden,June2003 Prof.Dr.GerhardMusiol Prof.Dr.HeinerMu¨hlig Co-Authors Somechaptersorsectionsaretheresultofcooperationwithco-authors. Chapterresp. section Co-author SphericalTrigonometry(3.4.1–3.4.3.3) Dr.H.Nickel ,Dresden SphericalCurves(3.4.3.4) Prof.L.Marsolek,Berlin GeometricTransformations,CoordinateTrans- formations,PlanarProjections(3.5.4,3.5.5) Dr.I.Steinert,Du¨sseldorf QuaternionsandApplications(4.4) PDDr.S.Bernstein,Freiberg(Sachsen) ∗SeePrefacetotheFirstRussianEdition VII Logic(5.1),SetTheory(5.2),ClassicalAlgebraic Structures(5.3),ApplicationsofGroups (beyond)5.3.4,5.3.5.4–5.3.5.6),RingsandFields (5.3.7),VectorSpaces(5.3.8),BooleanAlgebra andSwitchAlgebra(5.7),UniversalAlgebra(5.6) Dr.J.Brunner,Dresden GroupRepresentation(5.3.4),Applicationsof Groups(5.3.5.4–5.3.5.6) Prof.Dr.R.Reif,Dresden Lie–GroupsandLie–Algebras(5.3.6) PDDr.S.Bernstein,Freiberg(Sachsen) ElementaryNumberTheory(5.4),Cryptology, (5.5)Graphs(5.8) Prof.Dr.U.Baumann,Dresden Fuzzy–Logic(5.9) Prof.Dr.A.Grauel,Soest ImportantFormulasfortheSphericalBessel Functions(9.1.2.6,sub-point2.5) Prof.Dr.P.Ziesche,Dresden StatisticalInterpretationoftheWaveFunction (9.2.4.4) Prof.Dr.R.Reif,Dresden Non-linearpartialDifferentialEquations: Solitons,PeriodicPatternsandChaos(9.2.5) Prof.Dr.P.Ziesche,Dresden DissipativeSolitons,LightandDark Solitons(9.2.5.3,inpoint2) Dr.J.Brand,Dresden LinearIntegralEquations(11) Dr.I.Steinert,Du¨sseldorf Functionalanalysis(12) Prof.Dr.M.Weber,Dresden EllipticFunctions(14.6) Dr.N.M.Fleischer ,Moskau DynamicalSystemsandChaos(17) Prof.Dr.V.Reitmann,St.Petersburg Optimization(18) Dr.I.Steinert,Du¨sseldorf UsingtheComputer:(19.8.1,19.8.2),Interactive System:Mathematica(19.8.4.2),Maple(19.8. 4.3),ComputeralgebraSystems–ExampleMathe- matica(20) Prof.Dr.G.Flach,Dresden InteractiveSystem:Matlab(19.8.4.1) PDDr.B.Mulansky,Clausthal ComputeralgebraSystems–ExampleMathemati- ca(20):Revisionofthechapterinaccordancewith version10ofMathematica Dr.J.T´oth,Budapest AdditionalChapterswithCo-AuthorsintheCD–ROM totheBooksoftheGermanEditions7,8and9. Lie–GroupsandLie–Algebras(5.3.5),(5.3.6) Prof.Dr.R.Reif,Dresden Non-linearPartialDifferentialEquations: InverseScatteringTheory(methodsinanalogy totheFouriermethod)(9.2.6) Dr.B.Rumpf, MathematicalBasisofQuantumMechanics(21) Prof.Dr.A.Buchleitner,PDDr.M.Tiersch, Dr.Th.Wellens,Freiburg Quantencomputer(22) Prof.Dr.A.Buchleitner,PDDr.M.Tiersch, Dr.Th.Wellens,Freiburg VIII Contents Contents ListofTables XLII 1 Arithmetics 1 1.1 ElementaryRulesforCalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1.1 Natural,Integer,andRationalNumbers. . . . . . . . . . . . . . . 1 1.1.1.2 IrrationalandTranscendentalNumbers . . . . . . . . . . . . . . . 2 1.1.1.3 RealNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1.4 ContinuedFractions . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1.5 Commensurability . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 MethodsforProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2.1 DirectProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2.2 IndirectProoforProofbyContradiction . . . . . . . . . . . . . . 5 1.1.2.3 MathematicalInduction . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2.4 ConstructiveProof . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 SumsandProducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3.1 Sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3.2 Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.4 Powers,Roots,andLogarithms. . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.4.1 Powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.4.2 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.4.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.4.4 SpecialLogarithms . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.5 AlgebraicExpressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.5.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.5.2 AlgebraicExpressionsinDetail . . . . . . . . . . . . . . . . . . . 11 1.1.6 IntegralRationalExpressions. . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.6.1 RepresentationinPolynomialForm . . . . . . . . . . . . . . . . . 11 1.1.6.2 FactoringPolynomials . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.6.3 SpecialFormulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.6.4 BinomialTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.6.5 DeterminationoftheGreatestCommonDivisorofTwoPolynomials 14 1.1.7 RationalExpressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.7.1 ReducingtotheSimplestForm. . . . . . . . . . . . . . . . . . . . 14 1.1.7.2 DeterminationoftheIntegralRationalPart. . . . . . . . . . . . . 15 1.1.7.3 PartialFractionDecomposition . . . . . . . . . . . . . . . . . . . 15 1.1.7.4 TransformationsofProportions . . . . . . . . . . . . . . . . . . . 17 1.1.8 IrrationalExpressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 FiniteSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.1 DefinitionofaFiniteSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.2 ArithmeticSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.3 GeometricSeries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.4 SpecialFiniteSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.5 MeanValues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.5.1 ArithmeticMeanorArithmeticAverage . . . . . . . . . . . . . . 19 1.2.5.2 GeometricMeanorGeometricAverage . . . . . . . . . . . . . . . 20 1.2.5.3 HarmonicMean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.5.4 QuadraticMean. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Contents IX 1.2.5.5 RelationsBetweentheMeansofTwoPositiveValues. . . . . . . . 20 1.3 BusinessMathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.1 CalculationofInterestorPercentage. . . . . . . . . . . . . . . . . . . . . . 21 1.3.1.1 PercentageorInterest . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.1.2 Increment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.1.3 DiscountorReduction . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.2 CalculationofCompoundInterest . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.2.1 Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.2.2 CompoundInterest . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.3 AmortizationCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.3.1 Amortization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.3.2 EqualPrincipalRepayments . . . . . . . . . . . . . . . . . . . . . 23 1.3.3.3 EqualAnnuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.4 AnnuityCalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.4.1 Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.4.2 FutureAmountofanOrdinaryAnnuity . . . . . . . . . . . . . . . 25 1.3.4.3 Balanceaftern AnnuityPayments . . . . . . . . . . . . . . . . . 25 1.3.5 Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.5.1 MethodsofDepreciation . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.5.2 Straight-LineMethod. . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.5.3 ArithmeticallyDecliningBalanceDepreciation . . . . . . . . . . . 26 1.3.5.4 DigitalDecliningBalanceDepreciation . . . . . . . . . . . . . . . 27 1.3.5.5 GeometricallyDecliningBalanceDepreciation . . . . . . . . . . . 27 1.3.5.6 DepreciationwithDifferentTypesofDepreciationAccount . . . . 28 1.4 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.1 PureInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.1.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.1.2 PropertiesofInequalitiesofTypeIandII . . . . . . . . . . . . . . 29 1.4.2 SpecialInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.2.1 TriangleInequalityforRealNumbers . . . . . . . . . . . . . . . . 30 1.4.2.2 TriangleInequalityforComplexNumbers . . . . . . . . . . . . . . 30 1.4.2.3 InequalitiesforAbsoluteValuesofDifferencesofRealandComplex Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.2.4 InequalityforArithmeticandGeometricMeans . . . . . . . . . . 30 1.4.2.5 InequalityforArithmeticandQuadraticMeans. . . . . . . . . . . 30 1.4.2.6 InequalitiesforDifferentMeansofRealNumbers . . . . . . . . . . 30 1.4.2.7 Bernoulli’sInequality . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.2.8 BinomialInequality. . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.4.2.9 Cauchy-SchwarzInequality. . . . . . . . . . . . . . . . . . . . . . 31 1.4.2.10 ChebyshevInequality . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.4.2.11 GeneralizedChebyshevInequality . . . . . . . . . . . . . . . . . . 32 1.4.2.12 H¨olderInequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4.2.13 MinkowskiInequality . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4.3 SolutionofLinearandQuadraticInequalities . . . . . . . . . . . . . . . . . 33 1.4.3.1 GeneralRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4.3.2 LinearInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4.3.3 QuadraticInequalities . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4.3.4 GeneralCaseforInequalitiesofSecondDegree . . . . . . . . . . . 33 1.5 ComplexNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.1 ImaginaryandComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.1.1 ImaginaryUnit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.1.2 ComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 34 X Contents 1.5.2 GeometricRepresentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.2.1 VectorRepresentation . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.2.2 EqualityofComplexNumbers . . . . . . . . . . . . . . . . . . . . 35 1.5.2.3 TrigonometricFormofComplexNumbers. . . . . . . . . . . . . . 35 1.5.2.4 ExponentialFormofaComplexNumber . . . . . . . . . . . . . . 36 1.5.2.5 ConjugateComplexNumbers . . . . . . . . . . . . . . . . . . . . 36 1.5.3 CalculationwithComplexNumbers . . . . . . . . . . . . . . . . . . . . . . 36 1.5.3.1 AdditionandSubtraction . . . . . . . . . . . . . . . . . . . . . . 36 1.5.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.5.3.3 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.5.3.4 GeneralRulesfortheBasicOperations . . . . . . . . . . . . . . . 37 1.5.3.5 TakingPowersofComplexNumbers. . . . . . . . . . . . . . . . . 38 1.5.3.6 Takingthen-thRootofaComplexNumber . . . . . . . . . . . . 38 1.6 AlgebraicandTranscendentalEquations. . . . . . . . . . . . . . . . . . . . . . . . 38 1.6.1 TransformingAlgebraicEquationstoNormalForm . . . . . . . . . . . . . . 38 1.6.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.6.1.2 SystemsofnAlgebraicEquations . . . . . . . . . . . . . . . . . . 39 1.6.1.3 ExtraneousRoots . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.6.2 EquationsofDegreeatMostFour . . . . . . . . . . . . . . . . . . . . . . . 39 1.6.2.1 EquationsofDegreeOne(LinearEquations) . . . . . . . . . . . . 39 1.6.2.2 EquationsofDegreeTwo(QuadraticEquations) . . . . . . . . . . 40 1.6.2.3 EquationsofDegreeThree(CubicEquations) . . . . . . . . . . . 40 1.6.2.4 EquationsofDegreeFour. . . . . . . . . . . . . . . . . . . . . . . 42 1.6.2.5 EquationsofHigherDegree . . . . . . . . . . . . . . . . . . . . . 43 1.6.3 EquationsofDegreen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.6.3.1 GeneralPropertiesofAlgebraicEquations . . . . . . . . . . . . . 43 1.6.3.2 EquationswithRealCoefficients . . . . . . . . . . . . . . . . . . . 44 1.6.4 ReducingTranscendentalEquationstoAlgebraicEquations . . . . . . . . . 45 1.6.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.6.4.2 ExponentialEquations . . . . . . . . . . . . . . . . . . . . . . . . 46 1.6.4.3 LogarithmicEquations . . . . . . . . . . . . . . . . . . . . . . . . 46 1.6.4.4 TrigonometricEquations . . . . . . . . . . . . . . . . . . . . . . . 46 1.6.4.5 EquationswithHyperbolicFunctions . . . . . . . . . . . . . . . . 47 2 Functions 48 2.1 NotionofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.1 DefinitionofaFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.1.1 Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.1.2 RealFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.1.3 FunctionsofSeveralVariables . . . . . . . . . . . . . . . . . . . . 48 2.1.1.4 ComplexFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.1.5 FurtherFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.1.6 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.1.7 FunctionsandMappings . . . . . . . . . . . . . . . . . . . . . . . 49 2.1.2 MethodsforDefiningaRealFunction . . . . . . . . . . . . . . . . . . . . . 49 2.1.2.1 DefiningaFunction . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.1.2.2 AnalyticRepresentationofaFunction. . . . . . . . . . . . . . . . 49 2.1.3 CertainTypesofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.1.3.1 MonotoneFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.1.3.2 BoundedFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.1.3.3 ExtremeValuesofFunctions . . . . . . . . . . . . . . . . . . . . . 51 2.1.3.4 EvenFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51